The Encyclopedia of Mathematical and Philosophical Proofs

Welcome to our comprehensive collection of mathematical and philosophical proofs. This wiki aims to be a definitive resource for students, scholars, and enthusiasts seeking rigorous demonstrations across the spectrum of mathematical disciplines and philosophical traditions.

This encyclopedia serves multiple purposes:

• To collect and preserve important proofs in accessible formats
• To provide clear explanations with appropriate levels of rigor
• To highlight connections between different areas of mathematics and philosophy
• To trace the historical development of key theorems and arguments
• To serve as an educational resource for learners at various levels

• DokuWiki syntax guide
• Mathematical expression formatting
• How to navigate this encyclopedia
• Understanding our difficulty ratings
• Glossary of terms

• Proof formatting standards
• How to contribute new proofs
• Our peer review process
• Citation guidelines
• Tips for clear mathematical writing

• Logic and Set Theory
  • Gödel's Incompleteness Theorems
  • Compactness Theorem
  • Completeness Theorem
  • Löwenheim-Skolem Theorems
  • ZFC Axioms and Consequences
  • Axiom of Choice and Equivalents
  • Constructive Logic
• Number Systems
  • Peano Axioms
  • Construction of Integers
  • Construction of Rational Numbers
  • Construction of Real Numbers
  • Complex Numbers
  • Cardinal Numbers
  • Ordinal Numbers

• Elementary Number Theory
  • Division Algorithm
  • Euclidean Algorithm
  • Fundamental Theorem of Arithmetic
  • Fermat's Little Theorem
  • Euler's Theorem
  • Chinese Remainder Theorem
• Analytic Number Theory
  • Prime Number Theorem
  • Dirichlet's Theorem on Primes in Arithmetic Progressions
• Algebraic Number Theory
  • Quadratic Reciprocity Law
  • Dirichlet's Unit Theorem
  • Class Number Formula

• Group Theory
  • Lagrange's Theorem
  • Sylow Theorems
  • Isomorphism Theorems
  • Classification of Finite Simple Groups
• Ring Theory
  • Ideal Theory
  • Unique Factorization Domains
  • Principal Ideal Domains
• Field Theory
  • Field Extensions
  • Splitting Fields
  • Algebraic Closure
• Galois Theory
  • Fundamental Theorem of Galois Theory
  • Solvability by Radicals
  • Unsolvability of the Quintic
• Linear Algebra
  • Determinants
  • Eigenvalues and Eigenvectors
  • Diagonalization
  • Jordan Normal Form
  • Spectral Theorem

• Calculus
  • Limit Theorems
  • Continuity
  • Differentiation Rules
  • Mean Value Theorem
  • Integration Techniques
  • Fundamental Theorem of Calculus
• Real Analysis
  • Convergence of Sequences
  • Convergence of Series
  • Continuity Properties
  • Differentiability Theorems
  • Riemann Integration
  • Lebesgue Integration
• Complex Analysis
  • Cauchy's Integral Theorem
  • Residue Theorem
  • Laurent Series
  • Liouville's Theorem
  • Maximum Modulus Principle
• Functional Analysis
  • Baire Category Theorem
  • Uniform Boundedness Principle
  • Open Mapping Theorem
  • Closed Graph Theorem
  • Hahn-Banach Theorem
  • Spectral Theory

• Euclidean Geometry
  • Pythagorean Theorem
  • Triangle Congruence Theorems
  • Similar Triangles
  • Circle Theorems
• Non-Euclidean Geometry
  • Hyperbolic Geometry
  • Elliptic Geometry
• Differential Geometry
  • Theory of Curves
  • Theory of Surfaces
  • Manifolds
  • Gauss-Bonnet Theorem
• Point-Set Topology
  • Metric Spaces
  • Continuous Functions
  • Compactness
  • Connectedness
• Algebraic Topology
  • Fundamental Group
  • Covering Spaces
  • Homology Groups
  • Cohomology Theories
  • Fixed Point Theorems

• Probability Theory
  • Kolmogorov Axioms
  • Conditional Probability
  • Probability Distributions
  • Limit Theorems
• Statistics
  • Parameter Estimation
  • Hypothesis Testing
  • Regression Analysis
• Differential Equations
  • First-Order ODEs
  • Linear ODEs
  • Systems of ODEs
  • Partial Differential Equations
• Numerical Analysis
  • Interpolation
  • Numerical Integration
  • Optimization Methods

• Fermat's Last Theorem
• Four Color Theorem
• Poincaré Conjecture
• Riemann Hypothesis
• P vs NP Problem
• Goldbach Conjecture
• Twin Prime Conjecture

• Ontological Arguments
  • Anselm's Ontological Argument
  • Descartes' Ontological Argument
  • Modal Ontological Arguments
  • Plantinga's Ontological Argument
• Causation
  • Cosmological Arguments
  • Kalam Cosmological Argument
  • Leibniz's Cosmological Argument
  • Infinite Regress Arguments
• Philosophy of Mind
  • Arguments for Dualism
  • Arguments for Physicalism
  • Explanatory Gap Arguments
  • Knowledge Argument (Mary's Room)
  • Zombie Arguments

• Nature of Knowledge
  • Knowledge as Justified True Belief
  • Gettier Problems
  • Reliabilist Theories
  • Coherentist Theories
  • Foundationalist Theories
• Skeptical Arguments
  • Brain in a Vat
  • Descartes' Evil Demon
  • Dream Argument
  • Responses to Skepticism
• Problem of Induction
  • Hume's Problem of Induction
  • Goodman's New Riddle of Induction

• Normative Ethics
  • Utilitarian Arguments
  • Kantian Arguments
  • Virtue Ethics
  • Natural Law Theory
• Metaethics
  • Moral Realism
  • Error Theory
  • Emotivism
  • Prescriptivism
• Applied Ethics
  • Trolley Problem
  • Value of Life Arguments
  • Implications of Consequentialism

• Formal Logic
  • Propositional Logic
  • Predicate Logic
  • Modal Logic
  • Non-Classical Logics
• Informal Logic and Fallacies
  • Deductive Fallacies
  • Inductive Fallacies
  • Relevance Fallacies
• Theories of Meaning
  • Reference Theories
  • Use Theories
  • Verification Principle
  • Gricean Pragmatics

• Demarcation Problem
  • Popper's Falsificationism
  • Kuhn's Paradigm Shifts
• Scientific Explanation
  • Deductive-Nomological Model
  • Causal Accounts
• Scientific Realism vs. Anti-Realism
  • No Miracles Argument
  • Pessimistic Induction

• Philosophical Foundations of Mathematics
• Mathematical Platonism
• Formalism
• Intuitionism
• Structuralism

• Mathematical Physics
• Mathematical Biology
• Mathematical Models in Social Sciences

• Undergraduate Mathematics
• Graduate Mathematics
• Philosophy Courses

• Mathematical Notation Guide
• Symbol Reference
• Glossary of Terms
• Comprehensive Bibliography

• Mathematical Proof Template
• Philosophical Argument Template
• Reference and Citation Template
• Theorem Template
• Lemma Template

• 2025-03-15 - Wiki structure established
• 2025-03-15 - Initial proof templates created
• 2025-03-15 - Mathematical expression formatting guide added

We welcome contributions from mathematicians, philosophers, educators, and enthusiasts. To contribute:

• Follow our standards: Review our proof formatting standards
• Use proper notation: Ensure mathematical expressions use LaTeX via the MathJax plugin
• Provide references: Include sources for all theorems, proofs, and arguments
• Maintain organization: Follow our established namespace structure
• Consider accessibility: Write clear explanations that build on prerequisites
• Add value: Provide historical context, alternative proofs, and applications

For substantial contributions, please:

• Contact an editor to discuss the addition
• Submit drafts for peer review
• Be open to feedback and revisions

This encyclopedia is maintained by an editorial team of mathematicians and philosophers. For questions, suggestions, or to join the editorial team, please contact the administrators.

For questions or assistance with this wiki, please contact ayman@aymanlafaz.com.